Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscrngo Structured version   Visualization version   GIF version

Theorem iscrngo 33413
 Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Assertion
Ref Expression
iscrngo (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Proof of Theorem iscrngo
StepHypRef Expression
1 df-crngo 33411 . 2 CRingOps = (RingOps ∩ Com2)
21elin2 3784 1 (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1992  RingOpscrngo 33311  Com2ccm2 33406  CRingOpsccring 33410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-in 3567  df-crngo 33411 This theorem is referenced by:  iscrngo2  33414  iscringd  33415  crngorngo  33417  fldcrng  33421  isfld2  33422  isdmn2  33472
 Copyright terms: Public domain W3C validator